In working out why the absolute value can be ommitted and what you meant by trivial. I seen an argument that doesn't require induction. If you're interested: Assuming m = n + k,

Group the terms as follows (for even k)

(and for odd k)

Then we have a series of positive terms subtracted from so

Thanks for your input

I think it works just fine! Yet even in this case we have what's sometimes called "concealed or hidden" induction: it is a weak form and it just could be stressed that what you did actually works no matter what in you used...